Choosing axes means deciding which two calculated quantities to plot, then filling in a table for them. Once the two columns are right, plotting is routine.
This page continues from turning a curve into a straight line in the linear law chapter.
How do I pick the axes from the relation?
Start from the linear form. The expression that stands alone on the left is the vertical axis. The expression multiplied by the gradient is the horizontal axis.
Then add two new columns to the table the question gives you: one for Y and one for X. Calculate each row from the original x and y values.
Worked example: filling the table
The variables x and y are connected by y = hx + k/x, where h and k are constants. The table gives these values:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 5 | 5.5 | 7 | 8.75 | 10.6 |
Multiply through by x to remove the fraction: xy = hx² + k. This has the form Y = mX + c, where Y = xy and X = x².
Now calculate the new columns:
| X = x² | 1 | 4 | 9 | 16 | 25 |
|---|---|---|---|---|---|
| Y = xy | 5 | 11 | 21 | 35 | 53 |
Check the first and last points against the line Y = 2X + 3: 2(1) + 3 = 5 and 2(25) + 3 = 53. Both match, so the values are consistent with h = 2 and k = 3.
Labelling and scaling
Your axis labels must say exactly what you plotted. Write “xy” on the vertical axis and “x²” on the horizontal axis, not “y” and “x”.
For the scale, pick equal steps with easy values. Here Y runs from 5 to 53, so 10 units per large square suits the vertical axis. The X values run from 1 to 25, so 5 units per large square fits the horizontal axis.
The mistake that costs the marks
The common slip is to plot the original values of y against x and then draw a line through what is really a curve. A second slip is a correct table with axes labelled only “y” and “x”.
| Item | Wrong | Right |
|---|---|---|
| Points plotted | (1, 5), (2, 5.5), (3, 7) | (1, 5), (4, 11), (9, 21) |
| Vertical label | y | xy |
| Horizontal label | x | x² |
| Shape | Curve | Straight line |
Check yourself
The relation is y = px² + qx. A student plots y against x² and gets a curve. Which quantities should be plotted instead?
Answer
Divide every term by x: y/x = px + q.
So Y = y/x on the vertical axis and X = x on the horizontal axis. The gradient is p and the intercept is q.
The student plotted y against x², which would only work if the qx term were absent.
What to study next
With the points plotted and the line drawn, the next step is reading the constants. Continue with finding constants from gradient and intercept. The graph evidence comparison lab and the mistake log help you check and record slips.
If you want a teacher to watch your table and scale choices live, see online one-to-one Additional Mathematics tuition.